Cutting devices (such as disclosed in U.S. Pat. No. 5,267,817 and in WO 99/50,012) are obtainable in a plurality of conventional embodiments and are freely available on the market. The conventional approaches are used especially for end-cut turning and plain turning, and for cut-off operations on a rotating workpiece made generally rotationally symmetrical as a turned part. These known cutting devices have an elongated holder with a square cross section which can be fixed over the area of its one free end on a tool holder. Several cutting devices can be held in a holder device, often under constricted installation conditions. The cutting devices can be located horizontally in a row on top of one another in the overall holder device, but also directly next to one another. On its other end, the holder undergoes transition in one piece into a block-like clamping head which holds the metal-cutting tool on its side by way of a receptacle. This block-like clamping head has essentially planar surfaces to all sides which transition into one another at a right angle. In particular the front face of the clamping head is made in the form of a straight plane extending at a right angle to the bottom and top and the longitudinal sides of the holder. The metal-cutting tool with its machining cutting edge projects from this flat plane forward in the direction of the machining zone. With respect to the indicated geometrical structure of the clamping head, the cutting edge of the machining tool on the rotating workpiece must first create a clear space so that for deeper machining the clamping head can follow over its entire width. In particular, in a constricted installation situation for the cutting device and for cramped machining areas on the rotating workpiece into which, for example, only a thin machining groove is to be made, the known approaches are unsuitable. In these instances the rotating workpiece would then collide with the front face of the clamping head.